A high-precision calculation of the electron anomalous magnetic moment requires an evaluation of QED Feynman diagrams up to five independent loops. To make this calculation practically feasible it is necessary to remove all infrared and ultraviolet divergences before integration. A procedure of removing both infrared and ultraviolet divergences in each individual Feynman diagram will be presented. The procedure is based on linear operators that are applied to the Feynman amplitudes of ultraviolet divergent subdiagrams. The usage of linear operators allows us to avoid residual renormalizations after subtraction of divergences. This procedure leads immediately to finite Feynman parametric integrals. A method of Monte Carlo integration of these Feynman parametric integrands will be presented. The method is based on importance sampling. The probability density function is constructed for each Feynman diagram individually by using some combinatorial information from the diagram. The calculated value of the total contribution of the 5-loop QED Feynman diagrams without lepton loops to the electron anomalous magnetic moment will be presented. This result was obtained by a GPU-based computation on a supercomputer. The calculation provides double-checking of the value. The contributions of nine gauge-invariant classes of 5-loop Feynman diagrams without lepton loops will be presented for the first time. Also, the contributions of some individual 6-loop Feynman diagrams will be given for demonstration of the method.